Problem : Find a vector which is perpendicular to both u = (3, 0, 2) and v = (1, 1, 1).

The vector u×v is perpendicular to both u and v, so we need only compute the cross product to do this problem. From the component formula, u×v = (- 2, - 1, 3). Using the dot product we can check that this vector is indeed perpendicular to u and v.

Problem : A triangle has two sides of length 5 and 6. If the triangle's area is 12, what is the angle between these two sides?

The given sides of the triangle can be thought of as two vectors u and v with magnitudes 5 and 6, respectively. From the geometric formula for the cross product, we know that the area of the parallelogram defined by these vectors is given by A = | u|| v| sinθ = 30 sinθ. The area A of the parallelogram is exactly twice the area of the triangle in question. Hence we can solve for sinθ = 24/30 = 4/5.

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